What is the future value of $100 at 10 percent simple interest for 2 years?
|
The figures in the table are easily calculated by multiplying the previous year’s value by 1.10, 1 representing the principal value and .10 representing the interest rate expressed as a decimal. So $100 today (year = 0) is, at 10 percent interest compounded annually, worth $110 in a year (100 × 1.1), $121 after two years (110 × 1.1), $131.10 after three years (121 × 1.1), and so forth. The quick way to calculate this for any year is to use the following formula: Show
FV = PV(1 + i)n where FV = the future value (the value of your investment in the future) PV = the present value (the amount of your investment today) (1 + i)n = the future value factor (aka the present value factor or discount factor in the equation below) i = interest rate (decimalized, for example, 6% = .06; 25% = .25, 2.763% = .02763, etc.) n = number of terms (here, years; elsewhere days, months, quarters) For $100 borrowed today at 10 percent compounded annually, in 100 years I’d owe you $1,378,061 (FV = 100 × 1.1100). (Good luck collecting that one!) What if someone offers to pay you, say, $1,000 in 5 years? How much would you be willing to pay today for that? Clearly, something less than $1,000. Instead of taking a PV and expanding it via multiplication to determine an FV, here you must do the opposite, or in other words, reduce or “discount” an FV to a PV. You do so by dividing, as in the following formula: PV = FV/(1+i)nor PV = 1000/(1+i)5Obviously, we can’t solve this equation unless one of the two remaining variables is given. If the interest rate is given as 5 percent, you would pay $783.53 today for $1,000 payable in 5 years (PV = 1000/1.055). If it is 20 percent, you’d give only $401.88 (PV = 1000/1.25). If it is 1 percent, you would give $951.47 (PV = 1000/1.015). Notice that as the interest rate rises (falls), the present value (price) of the future payment falls (rises). In other words, the price (PV) of some future payment (some FV; generically, a bond) and the rate of interest are inversely related. You can see this algebraically by noting that the i term is in the denominator, so as it gets larger, PV must get smaller (holding FV constant, of course). Economically this makes sense because a higher interest rate means a higher opportunity cost for money, so a sum payable in the future is worth less the more dear money is (the more it costs to borrow it). If payment of the bond described just above were to be made in ten years instead of five, at 1 percent interest per year, you’d pay $905.29 (PV = 1000/1.0110). Note here that, holding the interest rate (and all other factors) constant, you give less today for a payment further in the future ($905.29 < $951.47). That too makes good sense because you’re without your money longer and need to be compensated for it by paying a lower price for the bond/promise/IOU today. Stop and Think BoxCongratulations, you just won the Powerball: $100 million payable in $5 million installments over 20 years! Did you really win $100 million? (Hint: Calculate the PV of the final payment with interest at 4 percent.) No; 5 × 20 = 100, but the money payable next year and in subsequent years is not worth $5 million today if interest rates are above 0, and they almost always are. For example, the last payment, with interest rates at 4 percent compounded annually, has a PV of only 5,000,000/(1.04)20 = $2,281,934.73. This is a great place to stop and drill until calculating present value and future value becomes second nature to you. Work through the following problems until it hurts. Then do them again, standing on your head or on one leg. ExercisesFor all questions in this set, interest compounds annually and there are no transaction fees, defaults, etc. The figures in the table are easily calculated by multiplying the previous year’s value by 1.10, 1 representing the principal value and .10 representing the interest rate expressed as a decimal. So $100 today (year = 0) is, at 10 percent interest compounded annually, worth $110 in a year (100 × 1.1), $121 after two years (110 × 1.1), $131.10 after three years (121 × 1.1), and so forth. The quick way to calculate this for any year is to use the following formula: Nội dung chính Show FV = PV(1 + i)n where FV = the future value (the value of your investment in the future) PV = the present value (the amount of your investment today) (1 + i)n = the future value factor (aka the present value factor or discount factor in the equation below) i = interest rate (decimalized, for example, 6% = .06; 25% = .25, 2.763% = .02763, etc.) n = number of terms (here, years; elsewhere days, months, quarters) For $100 borrowed today at 10 percent compounded annually, in 100 years I’d owe you $1,378,061 (FV = 100 × 1.1100). (Good luck collecting that one!) What if someone offers to pay you, say, $1,000 in 5 years? How much would you be willing to pay today for that? Clearly, something less than $1,000. Instead of taking a PV and expanding it via multiplication to determine an FV, here you must do the opposite, or in other words, reduce or “discount” an FV to a PV. You do so by dividing, as in the following formula: PV = FV/(1+i)nor PV = 1000/(1+i)5Obviously, we can’t solve this equation unless one of the two remaining variables is given. If the interest rate is given as 5 percent, you would pay $783.53 today for $1,000 payable in 5 years (PV = 1000/1.055). If it is 20 percent, you’d give only $401.88 (PV = 1000/1.25). If it is 1 percent, you would give $951.47 (PV = 1000/1.015). Notice that as the interest rate rises (falls), the present value (price) of the future payment falls (rises). In other words, the price (PV) of some future payment (some FV; generically, a bond) and the rate of interest are inversely related. You can see this algebraically by noting that the i term is in the denominator, so as it gets larger, PV must get smaller (holding FV constant, of course). Economically this makes sense because a higher interest rate means a higher opportunity cost for money, so a sum payable in the future is worth less the more dear money is (the more it costs to borrow it). If payment of the bond described just above were to be made in ten years instead of five, at 1 percent interest per year, you’d pay $905.29 (PV = 1000/1.0110). Note here that, holding the interest rate (and all other factors) constant, you give less today for a payment further in the future ($905.29 < $951.47). That too makes good sense because you’re without your money longer and need to be compensated for it by paying a lower price for the bond/promise/IOU today. Stop and Think BoxCongratulations, you just won the Powerball: $100 million payable in $5 million installments over 20 years! Did you really win $100 million? (Hint: Calculate the PV of the final payment with interest at 4 percent.) No; 5 × 20 = 100, but the money payable next year and in subsequent years is not worth $5 million today if interest rates are above 0, and they almost always are. For example, the last payment, with interest rates at 4 percent compounded annually, has a PV of only 5,000,000/(1.04)20 = $2,281,934.73. This is a great place to stop and drill until calculating present value and future value becomes second nature to you. Work through the following problems until it hurts. Then do them again, standing on your head or on one leg. ExercisesFor all questions in this set, interest compounds annually and there are no transaction fees, defaults, etc. This compound interest calculator is a tool to help you estimate how much money you will earn on your deposit. In order to make smart financial decisions, you need to be able to foresee the final result. That's why it's worth knowing how to calculate compound interest. The most common real-life application of the compound interest formula is a regular savings calculation. Nội dung chính Show
Read on to find answers to the following questions:
You may also want to check our student loan calculator where you can make a projection on your expenses and study the effect of different student loan options on your budget. Interest rate definitionIn finance, interest rate is defined as the amount charged by a lender to a borrower for the use of an asset. So, for the borrower the interest rate is the cost of the debt, while for the lender it is the rate of return. Note that in the case where you make a deposit into a bank (e.g., put money in your savings account), you have, from a financial perspective, lent money to the bank. In such a case the interest rate reflects your profit. The interest rate is commonly expressed as a percentage of the principal amount (outstanding loan or value of deposit). Usually, it is presented on an annual basis, which is known as the annual percentage yield (APY) or effective annual rate (EAR). What is the compound interest definition?Generally, compound interest is defined as interest that is earned not solely on the initial amount invested but also on any further interest. In other words, compound interest is the interest on both the initial principal and the interest which has been accumulated on this principle so far. Therefore, the fundamental characteristic of compound interest is that interest itself earns interest. This concept of adding a carrying charge makes a deposit or loan grow at a faster rate. You can use the compound interest equation to find the value of an investment after a specified period or estimate the rate you have earned when buying and selling some investments. It also allows you to answer some other questions, such as how long it will take to double your investment. We will answer these questions in the examples below. Simple vs. compound interestYou should know that simple interest is something different than the compound interest. It is calculated only on the initial sum of money. On the other hand, compound interest is the interest on the initial principal plus the interest which has been accumulated. Compounding frequencyMost financial advisors will tell you that the compound frequency is the compounding periods in a year. But if you are not sure what compounding is, this definition will be meaningless to you… To understand this term you should know that compounding frequency is an answer to the question How often is the interest added to the principal each year? In other words, compounding frequency is the time period after which the interest will be calculated on top of the initial amount. For example:
Note that the greater the compounding frequency is, the greater the final balance. However, even when the frequency is unusually high, the final value can't rise above a particular limit. To understand the math behind this, check out our natural logarithm calculator. As the main focus of the calculator is the compounding mechanism, we designed a chart where you can follow the progress of the annual interest balances visually. If you choose a higher than yearly compounding frequency, the diagram will display the resulting extra or additional part of interest gained over yearly compounding by the higher frequency. Thus, in this way, you can easily observe the real power of compounding. Compound interest formulaThe compound interest formula is an equation that lets you estimate how much you will earn with your savings account. It's quite complex because it takes into consideration not only the annual interest rate and the number of years but also the number of times the interest is compounded per year. The formula for annual compound interest is as follows:
Where:
It is worth knowing that when the compounding period is one ( How to calculate compound interestActually, you don't need to memorize the compound interest formula from the previous section to estimate the future value of your investment. In fact, you don't even need to know how to calculate compound interest! Thanks to our compound interest calculator you can do it in just a few seconds, whenever and wherever you want. (NB: Have you already tried the mobile version of our calculators?) With our smart calculator, all you need to calculate the future value of your investment is to fill the appropriate fields:
That's it! In a flash, our compound interest calculator makes all necessary computations for you and gives you the results. The two main results are:
In case you set the additional deposit field, we gave you the results for the compounded initial balance and compounded additional balance. Besides, we also show you their contribution to the total interest amount, namely, interest on the initial balance and interest on the additional deposit. Compound interest examples
The following examples are there to try and help you answer these questions. We believe that after studying them, you won't have any trouble with the understanding and practical implementation of compound interest. Example 1 – basic calculation of the value of an investmentThe first example is the simplest, in which we calculate the future value of an initial investment. Question You invest $10,000 for 10 years at the annual interest rate of 5%. The interest rate is compounded yearly. What will be the value of your investment after 10 years? Solution Firstly let’s determine what values are given, and what we need to find. We know that you are going to invest We want to calculate the amount of money you will receive from this investment, that is, we want to find the future value To count it, we need to plug in the appropriate numbers into the compound interest formula:
Answer The value of your investment after 10 years will be $16,288.95. Your profit will be Note that when doing calculations you must be very careful with your rounding. You shouldn't do too much until the very end. Otherwise, your answer may be incorrect. The accuracy is dependent on the values you are computing. For standard calculations, six digits after the decimal point should be enough. Example 2 - complex calculation of the value of an investmentIn the second example, we calculate the future value of an initial investment in which interest is compounded monthly. Question You invest $10,000 at the annual interest rate of 5%. The interest rate is compounded monthly. What will be the value of your investment after 10 years? Solution Like in the first example, we should determine the values first. The initial balance Let's plug in the appropriate numbers in the compound interest formula:
Answer The value of your investment after 10 years will be $16,470.09. Your profit will be Did you notice that this example is quite similar to the first one? Actually, the only difference is the compounding frequency. Note that, only thanks to more frequent compounding this time you will earn $181.14 more during the same period! ( Example 3 - Calculating the interest rate of an investment using the compound interest formulaNow, let's try a different type of question that can be answered using the compound interest formula. This time, some basic algebra transformations will be required. In this example, we will consider a situation in which we know the initial balance, final balance, number of years and compounding frequency but we are asked to calculate the interest rate. This type of calculation may be applied in a situation where you want to determine the rate earned when buying and selling an asset (e.g., property) which you are using as an investment. Data and question Solution Let's try to plug this numbers in the basic compound interest formula:
So:
We can solve this equation using the following steps:
Raise both sides to the 1/6th power
Subtract 1 from both sides
Finally solve for r
Answer In this example you earned $1,000 out of the initial investment of $2,000 within the six years, meaning that your annual rate was equal to 6.9913%. As you can see this time, the formula is not very simple and requires a lot of calculations. That's why it's worth testing our compound interest calculator, which solves the same equations in an instant, saving you time and effort. Example 4 - Calculating the doubling time of an investment using the compound interest formulaHave you ever wondered how many years it will take for your investment to double its value? Besides its other capabilities, our calculator can help you to answer this question. To understand how it does it, let's take a look at the following example. Data and question You put $1,000 on your saving account. Assuming that the interest rate is equal to 4% and it is compounded yearly. Find the number of years after which the initial balance will double. Solution The given values are as follows: the initial balance Let's start with the basic compound interest equation:
Knowing that
Which could be written as
Divide both sides by P (P mustn't be 0!)
To solve for t, you need take the natural log (ln), of both sides:
So
Answer In our example it takes 18 years (18 is the nearest integer that is higher than 17.67) to double the initial investment. Have you noticed that in the above solution we didn't even need to know the initial and final balances of the investment? It is thanks to the simplification we made in the third step (Divide both sides by P). However, when using our compound interest rate calculator, you will need to provide this information in the appropriate fields. Don't worry if you just want to find the time in which the given interest rate would double your investment, just type in any numbers (for example It is also worth knowing that exactly the same calculations may be used to compute when the investment would triple (or multiply by any number in fact). All you need to do is just use a different multiple of P in the second step of the above example. You can also do it with our calculator. Compound interest tableCompound interest tables were used everyday, before the era of calculators, personal computers, spreadsheets, and unbelievable solutions provided by Omni Calculator 😂. The tables were designed to make the financial calculations simpler and faster (yes, really…). They are included in many older financial textbooks as an appendix. Below, you can see what a compound interest table looks like. Using the data provided in the compound interest table you can calculate the final balance of your investment. All you need to know is that the column compound amount factor shows the value of the factor Note that the values from the column Present worth factor are used to compute the present value of the investment when you know its future value. Obviously, this is only a basic example of a compound interest table. In fact, they are usually much, much larger, as they contain more periods With your new knowledge of how the world of financial calculations looked before Omni Calculator, do you enjoy our tool? Why not share it with your friends? Let them know about Omni! If you want to be financially smart, you can also try our other finance calculators. Additional InformationNow that you know how to calculate compound interest, it's high time you found other applications to help you make the greatest profit from your investments: To compare bank offers which have different compounding periods, we need to calculate the Annual Percentage Yield, also called Effective Annual Rate (EAR). This value tells us how much profit we will earn within a year. The most comfortable way to figure it out is using the APY calculator, which estimates the EAR from the interest rate and compounding frequency. If you want to find out how long it would take for something to increase by n%, you can use our rule of 72 calculator. This tool enables you to check how much time you need to double your investment even quicker than the compound interest rate calculator. You may also be interested in the credit card payoff calculator, which allows you to estimate how long it will take until you are completely debt-free. Another interesting calculator is our cap rate calculator which determines the rate of return on your real estate property purchase. We also suggest you try the lease calculator which helps you determine the monthly and total payments for a lease. If you're looking to finance the purchase of a new recreational vehicle (RV), our RV loan calculator makes it simple to work out what the best deal will be for you. The depreciation calculator enables you to use three different methods to estimate how fast the value of your asset decreases over time. What is the FV of $100 in 1 year if the interest rate is 10% per year?For example, if you decided to invest $100.00 at an interest rate of 10% – assuming a compounding frequency of 1 – the investment should be worth $110 by the end of one year. However, if the interest compounds semi-annually, the investment is worth $121 instead.
What is the present value of $100 with the 10% interest rate if received one year from now?If the appropriate interest rate is 10 percent, then the present value of $100 spent or earned one year from now is $100 divided by 1.10, which is about $91.
What would the future value of 100 be after 5 years at 10% simple interest?Answer and Explanation: The $100 investment becomes $161.05 after 5 years at 10% compound interest.
What is the future value of $10000 on deposit for 2 years at 6% simple interest?The future value of $10,000 on deposit for 2 years at 6% simple interest is $11200.
|
